In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 2 + 3 cos t, y = 4 + 2 sin t; t = π

Verified step by step guidance

1. Identify the parametric equations given: \(x = 2 + 3 \cos t\) and \(y = 4 + 2 \sin t\).
2. Substitute the given value of the parameter \(t = \pi\) into the equation for \(x\): calculate \(x = 2 + 3 \cos \pi\).
3. Recall the value of \(\cos \pi\), which is \(-1\), and use it to simplify the expression for \(x\).
4. Substitute the same value \(t = \pi\) into the equation for \(y\): calculate \(y = 4 + 2 \sin \pi\).
5. Recall the value of \(\sin \pi\), which is \$0\(, and use it to simplify the expression for \)y\(. The coordinates of the point are then \)(x, y)$.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like ellipses or circles.

Evaluating Trigonometric Functions at Specific Angles

To find coordinates for a given parameter t, substitute t into the trigonometric functions (cos t and sin t). Knowing exact values of sine and cosine at common angles like π is essential for accurate calculation of points on the curve.

Coordinate Calculation from Parametric Form

Once the parameter value is substituted, calculate x and y by evaluating the expressions. This process converts the parametric form into a specific point (x, y) on the plane, representing the location on the curve at that parameter.